The Pentagon delivered a warning Tuesday about the growth of the Islamic State offshoot in Afghanistan, saying the branch has become “more operationally active” there by carving out territory and openly fighting the Taliban.

The Defense Department report also said violence in Afghanistan overall is on the rise and the security situation has worsened since June, with more effective attacks by insurgents and increased casualties among Afghan security forces. The DOD said advances by the Islamic State of Iraq and the Levan...